// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_REAL_SCHUR_H
#define EIGEN_REAL_SCHUR_H

#include "./HessenbergDecomposition.h"

namespace Eigen {

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class RealSchur
  *
  * \brief Performs a real Schur decomposition of a square matrix
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the
  * real Schur decomposition; this is expected to be an instantiation of the
  * Matrix class template.
  *
  * Given a real square matrix A, this class computes the real Schur
  * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
  * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
  * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
  * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
  * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
  * blocks on the diagonal of T are the same as the eigenvalues of the matrix
  * A, and thus the real Schur decomposition is used in EigenSolver to compute
  * the eigendecomposition of a matrix.
  *
  * Call the function compute() to compute the real Schur decomposition of a
  * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
  * constructor which computes the real Schur decomposition at construction
  * time. Once the decomposition is computed, you can use the matrixU() and
  * matrixT() functions to retrieve the matrices U and T in the decomposition.
  *
  * The documentation of RealSchur(const MatrixType&, bool) contains an example
  * of the typical use of this class.
  *
  * \note The implementation is adapted from
  * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
  * Their code is based on EISPACK.
  *
  * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
  */
template <typename _MatrixType> class RealSchur
{
public:
    typedef _MatrixType MatrixType;
    enum
    {
        RowsAtCompileTime = MatrixType::RowsAtCompileTime,
        ColsAtCompileTime = MatrixType::ColsAtCompileTime,
        Options = MatrixType::Options,
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };
    typedef typename MatrixType::Scalar Scalar;
    typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
    typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3

    typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
    typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;

    /** \brief Default constructor.
      *
      * \param [in] size  Positive integer, size of the matrix whose Schur decomposition will be computed.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via compute().  The \p size parameter is only
      * used as a hint. It is not an error to give a wrong \p size, but it may
      * impair performance.
      *
      * \sa compute() for an example.
      */
    explicit RealSchur(Index size = RowsAtCompileTime == Dynamic ? 1 : RowsAtCompileTime)
        : m_matT(size, size), m_matU(size, size), m_workspaceVector(size), m_hess(size), m_isInitialized(false), m_matUisUptodate(false), m_maxIters(-1)
    {
    }

    /** \brief Constructor; computes real Schur decomposition of given matrix. 
      * 
      * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
      * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
      *
      * This constructor calls compute() to compute the Schur decomposition.
      *
      * Example: \include RealSchur_RealSchur_MatrixType.cpp
      * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
      */
    template <typename InputType>
    explicit RealSchur(const EigenBase<InputType>& matrix, bool computeU = true)
        : m_matT(matrix.rows(), matrix.cols()), m_matU(matrix.rows(), matrix.cols()), m_workspaceVector(matrix.rows()), m_hess(matrix.rows()),
          m_isInitialized(false), m_matUisUptodate(false), m_maxIters(-1)
    {
        compute(matrix.derived(), computeU);
    }

    /** \brief Returns the orthogonal matrix in the Schur decomposition. 
      *
      * \returns A const reference to the matrix U.
      *
      * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
      * member function compute(const MatrixType&, bool) has been called before
      * to compute the Schur decomposition of a matrix, and \p computeU was set
      * to true (the default value).
      *
      * \sa RealSchur(const MatrixType&, bool) for an example
      */
    const MatrixType& matrixU() const
    {
        eigen_assert(m_isInitialized && "RealSchur is not initialized.");
        eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
        return m_matU;
    }

    /** \brief Returns the quasi-triangular matrix in the Schur decomposition. 
      *
      * \returns A const reference to the matrix T.
      *
      * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
      * member function compute(const MatrixType&, bool) has been called before
      * to compute the Schur decomposition of a matrix.
      *
      * \sa RealSchur(const MatrixType&, bool) for an example
      */
    const MatrixType& matrixT() const
    {
        eigen_assert(m_isInitialized && "RealSchur is not initialized.");
        return m_matT;
    }

    /** \brief Computes Schur decomposition of given matrix. 
      * 
      * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
      * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
      * \returns    Reference to \c *this
      *
      * The Schur decomposition is computed by first reducing the matrix to
      * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
      * matrix is then reduced to triangular form by performing Francis QR
      * iterations with implicit double shift. The cost of computing the Schur
      * decomposition depends on the number of iterations; as a rough guide, it
      * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
      * \f$10n^3\f$ flops if \a computeU is false.
      *
      * Example: \include RealSchur_compute.cpp
      * Output: \verbinclude RealSchur_compute.out
      *
      * \sa compute(const MatrixType&, bool, Index)
      */
    template <typename InputType> RealSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);

    /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
     *  \param[in] matrixH Matrix in Hessenberg form H
     *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
     *  \param computeU Computes the matriX U of the Schur vectors
     * \return Reference to \c *this
     * 
     *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
     *  using either the class HessenbergDecomposition or another mean. 
     *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
     *  When computeU is true, this routine computes the matrix U such that 
     *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
     * 
     * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
     * is not available, the user should give an identity matrix (Q.setIdentity())
     * 
     * \sa compute(const MatrixType&, bool)
     */
    template <typename HessMatrixType, typename OrthMatrixType>
    RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU);
    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was successful, \c NoConvergence otherwise.
      */
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "RealSchur is not initialized.");
        return m_info;
    }

    /** \brief Sets the maximum number of iterations allowed. 
      *
      * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
      * of the matrix.
      */
    RealSchur& setMaxIterations(Index maxIters)
    {
        m_maxIters = maxIters;
        return *this;
    }

    /** \brief Returns the maximum number of iterations. */
    Index getMaxIterations() { return m_maxIters; }

    /** \brief Maximum number of iterations per row.
      *
      * If not otherwise specified, the maximum number of iterations is this number times the size of the
      * matrix. It is currently set to 40.
      */
    static const int m_maxIterationsPerRow = 40;

private:
    MatrixType m_matT;
    MatrixType m_matU;
    ColumnVectorType m_workspaceVector;
    HessenbergDecomposition<MatrixType> m_hess;
    ComputationInfo m_info;
    bool m_isInitialized;
    bool m_matUisUptodate;
    Index m_maxIters;

    typedef Matrix<Scalar, 3, 1> Vector3s;

    Scalar computeNormOfT();
    Index findSmallSubdiagEntry(Index iu, const Scalar& considerAsZero);
    void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
    void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
    void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
    void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
};

template <typename MatrixType>
template <typename InputType>
RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU)
{
    const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)();

    eigen_assert(matrix.cols() == matrix.rows());
    Index maxIters = m_maxIters;
    if (maxIters == -1)
        maxIters = m_maxIterationsPerRow * matrix.rows();

    Scalar scale = matrix.derived().cwiseAbs().maxCoeff();
    if (scale < considerAsZero)
    {
        m_matT.setZero(matrix.rows(), matrix.cols());
        if (computeU)
            m_matU.setIdentity(matrix.rows(), matrix.cols());
        m_info = Success;
        m_isInitialized = true;
        m_matUisUptodate = computeU;
        return *this;
    }

    // Step 1. Reduce to Hessenberg form
    m_hess.compute(matrix.derived() / scale);

    // Step 2. Reduce to real Schur form
    // Note: we copy m_hess.matrixQ() into m_matU here and not in computeFromHessenberg
    //       to be able to pass our working-space buffer for the Householder to Dense evaluation.
    m_workspaceVector.resize(matrix.cols());
    if (computeU)
        m_hess.matrixQ().evalTo(m_matU, m_workspaceVector);
    computeFromHessenberg(m_hess.matrixH(), m_matU, computeU);

    m_matT *= scale;

    return *this;
}
template <typename MatrixType>
template <typename HessMatrixType, typename OrthMatrixType>
RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
{
    using std::abs;

    m_matT = matrixH;
    m_workspaceVector.resize(m_matT.cols());
    if (computeU && !internal::is_same_dense(m_matU, matrixQ))
        m_matU = matrixQ;

    Index maxIters = m_maxIters;
    if (maxIters == -1)
        maxIters = m_maxIterationsPerRow * matrixH.rows();
    Scalar* workspace = &m_workspaceVector.coeffRef(0);

    // The matrix m_matT is divided in three parts.
    // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
    // Rows il,...,iu is the part we are working on (the active window).
    // Rows iu+1,...,end are already brought in triangular form.
    Index iu = m_matT.cols() - 1;
    Index iter = 0;       // iteration count for current eigenvalue
    Index totalIter = 0;  // iteration count for whole matrix
    Scalar exshift(0);    // sum of exceptional shifts
    Scalar norm = computeNormOfT();
    // sub-diagonal entries smaller than considerAsZero will be treated as zero.
    // We use eps^2 to enable more precision in small eigenvalues.
    Scalar considerAsZero = numext::maxi<Scalar>(norm * numext::abs2(NumTraits<Scalar>::epsilon()), (std::numeric_limits<Scalar>::min)());

    if (norm != Scalar(0))
    {
        while (iu >= 0)
        {
            Index il = findSmallSubdiagEntry(iu, considerAsZero);

            // Check for convergence
            if (il == iu)  // One root found
            {
                m_matT.coeffRef(iu, iu) = m_matT.coeff(iu, iu) + exshift;
                if (iu > 0)
                    m_matT.coeffRef(iu, iu - 1) = Scalar(0);
                iu--;
                iter = 0;
            }
            else if (il == iu - 1)  // Two roots found
            {
                splitOffTwoRows(iu, computeU, exshift);
                iu -= 2;
                iter = 0;
            }
            else  // No convergence yet
            {
                // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
                Vector3s firstHouseholderVector = Vector3s::Zero(), shiftInfo;
                computeShift(iu, iter, exshift, shiftInfo);
                iter = iter + 1;
                totalIter = totalIter + 1;
                if (totalIter > maxIters)
                    break;
                Index im;
                initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
                performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
            }
        }
    }
    if (totalIter <= maxIters)
        m_info = Success;
    else
        m_info = NoConvergence;

    m_isInitialized = true;
    m_matUisUptodate = computeU;
    return *this;
}

/** \internal Computes and returns vector L1 norm of T */
template <typename MatrixType> inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
{
    const Index size = m_matT.cols();
    // FIXME to be efficient the following would requires a triangular reduxion code
    // Scalar norm = m_matT.upper().cwiseAbs().sum()
    //               + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
    Scalar norm(0);
    for (Index j = 0; j < size; ++j) norm += m_matT.col(j).segment(0, (std::min)(size, j + 2)).cwiseAbs().sum();
    return norm;
}

/** \internal Look for single small sub-diagonal element and returns its index */
template <typename MatrixType> inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& considerAsZero)
{
    using std::abs;
    Index res = iu;
    while (res > 0)
    {
        Scalar s = abs(m_matT.coeff(res - 1, res - 1)) + abs(m_matT.coeff(res, res));

        s = numext::maxi<Scalar>(s * NumTraits<Scalar>::epsilon(), considerAsZero);

        if (abs(m_matT.coeff(res, res - 1)) <= s)
            break;
        res--;
    }
    return res;
}

/** \internal Update T given that rows iu-1 and iu decouple from the rest. */
template <typename MatrixType> inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
{
    using std::abs;
    using std::sqrt;
    const Index size = m_matT.cols();

    // The eigenvalues of the 2x2 matrix [a b; c d] are
    // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
    Scalar p = Scalar(0.5) * (m_matT.coeff(iu - 1, iu - 1) - m_matT.coeff(iu, iu));
    Scalar q = p * p + m_matT.coeff(iu, iu - 1) * m_matT.coeff(iu - 1, iu);  // q = tr^2 / 4 - det = discr/4
    m_matT.coeffRef(iu, iu) += exshift;
    m_matT.coeffRef(iu - 1, iu - 1) += exshift;

    if (q >= Scalar(0))  // Two real eigenvalues
    {
        Scalar z = sqrt(abs(q));
        JacobiRotation<Scalar> rot;
        if (p >= Scalar(0))
            rot.makeGivens(p + z, m_matT.coeff(iu, iu - 1));
        else
            rot.makeGivens(p - z, m_matT.coeff(iu, iu - 1));

        m_matT.rightCols(size - iu + 1).applyOnTheLeft(iu - 1, iu, rot.adjoint());
        m_matT.topRows(iu + 1).applyOnTheRight(iu - 1, iu, rot);
        m_matT.coeffRef(iu, iu - 1) = Scalar(0);
        if (computeU)
            m_matU.applyOnTheRight(iu - 1, iu, rot);
    }

    if (iu > 1)
        m_matT.coeffRef(iu - 1, iu - 2) = Scalar(0);
}

/** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
template <typename MatrixType> inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
{
    using std::abs;
    using std::sqrt;
    shiftInfo.coeffRef(0) = m_matT.coeff(iu, iu);
    shiftInfo.coeffRef(1) = m_matT.coeff(iu - 1, iu - 1);
    shiftInfo.coeffRef(2) = m_matT.coeff(iu, iu - 1) * m_matT.coeff(iu - 1, iu);

    // Wilkinson's original ad hoc shift
    if (iter == 10)
    {
        exshift += shiftInfo.coeff(0);
        for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i, i) -= shiftInfo.coeff(0);
        Scalar s = abs(m_matT.coeff(iu, iu - 1)) + abs(m_matT.coeff(iu - 1, iu - 2));
        shiftInfo.coeffRef(0) = Scalar(0.75) * s;
        shiftInfo.coeffRef(1) = Scalar(0.75) * s;
        shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
    }

    // MATLAB's new ad hoc shift
    if (iter == 30)
    {
        Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
        s = s * s + shiftInfo.coeff(2);
        if (s > Scalar(0))
        {
            s = sqrt(s);
            if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
                s = -s;
            s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
            s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
            exshift += s;
            for (Index i = 0; i <= iu; ++i) m_matT.coeffRef(i, i) -= s;
            shiftInfo.setConstant(Scalar(0.964));
        }
    }
}

/** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
template <typename MatrixType>
inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
{
    using std::abs;
    Vector3s& v = firstHouseholderVector;  // alias to save typing

    for (im = iu - 2; im >= il; --im)
    {
        const Scalar Tmm = m_matT.coeff(im, im);
        const Scalar r = shiftInfo.coeff(0) - Tmm;
        const Scalar s = shiftInfo.coeff(1) - Tmm;
        v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im + 1, im) + m_matT.coeff(im, im + 1);
        v.coeffRef(1) = m_matT.coeff(im + 1, im + 1) - Tmm - r - s;
        v.coeffRef(2) = m_matT.coeff(im + 2, im + 1);
        if (im == il)
        {
            break;
        }
        const Scalar lhs = m_matT.coeff(im, im - 1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
        const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im - 1, im - 1)) + abs(Tmm) + abs(m_matT.coeff(im + 1, im + 1)));
        if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
            break;
    }
}

/** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
template <typename MatrixType>
inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
{
    eigen_assert(im >= il);
    eigen_assert(im <= iu - 2);

    const Index size = m_matT.cols();

    for (Index k = im; k <= iu - 2; ++k)
    {
        bool firstIteration = (k == im);

        Vector3s v;
        if (firstIteration)
            v = firstHouseholderVector;
        else
            v = m_matT.template block<3, 1>(k, k - 1);

        Scalar tau, beta;
        Matrix<Scalar, 2, 1> ess;
        v.makeHouseholder(ess, tau, beta);

        if (beta != Scalar(0))  // if v is not zero
        {
            if (firstIteration && k > il)
                m_matT.coeffRef(k, k - 1) = -m_matT.coeff(k, k - 1);
            else if (!firstIteration)
                m_matT.coeffRef(k, k - 1) = beta;

            // These Householder transformations form the O(n^3) part of the algorithm
            m_matT.block(k, k, 3, size - k).applyHouseholderOnTheLeft(ess, tau, workspace);
            m_matT.block(0, k, (std::min)(iu, k + 3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
            if (computeU)
                m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
        }
    }

    Matrix<Scalar, 2, 1> v = m_matT.template block<2, 1>(iu - 1, iu - 2);
    Scalar tau, beta;
    Matrix<Scalar, 1, 1> ess;
    v.makeHouseholder(ess, tau, beta);

    if (beta != Scalar(0))  // if v is not zero
    {
        m_matT.coeffRef(iu - 1, iu - 2) = beta;
        m_matT.block(iu - 1, iu - 1, 2, size - iu + 1).applyHouseholderOnTheLeft(ess, tau, workspace);
        m_matT.block(0, iu - 1, iu + 1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
        if (computeU)
            m_matU.block(0, iu - 1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
    }

    // clean up pollution due to round-off errors
    for (Index i = im + 2; i <= iu; ++i)
    {
        m_matT.coeffRef(i, i - 2) = Scalar(0);
        if (i > im + 2)
            m_matT.coeffRef(i, i - 3) = Scalar(0);
    }
}

}  // end namespace Eigen

#endif  // EIGEN_REAL_SCHUR_H
